Data driven consistency (working title)

We are motivated by applications that need rich model classes to represent them. Examples of rich model classes include distributions over large, countably infinite supports, slow mixing Markov processes, etc. But such rich classes may be too complex to admit estimators that converge to the truth with convergence rates that can be uniformly bounded over the entire model class as the sample size increases (uniform consistency). However, these rich classes may still allow for estimators with pointwise guarantees whose performance can be bounded in a model dependent way. The pointwise angle of course has the drawback that the estimator performance is a function of the very unknown model that is being estimated, and is therefore unknown. Therefore, even if the estimator is consistent, how well it is doing may not be clear no matter what the sample size is. Departing from the dichotomy of uniform and pointwise consistency, a new analysis framework is explored by characterizing rich model classes that may only admit pointwise guarantees, yet all the information about the model needed to guage estimator accuracy can be inferred from the sample at hand. To retain focus, we analyze the universal compression problem in this data driven pointwise consistency framework.